Probabilistic approach to the neumann problem
Pei Hsu · Communications on Pure and Applied Mathematics · 1985
Abstract The basic problem considered in this paper is to solve the following Neumann boundary value problem probabilistically: magnified image where we assume that q is in a certain functional class to be specified below, and φ is a bounded measurable function on the boundary. We give a martingale formulation of the Neumann problem and show that this formulation is essentially equivalent to the classical formulation. The paper culminates in an explicit formula for the solution of this problem in terms of reflecting Brownian motion and its boundary local time.